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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 2

Write Quadratics from Roots

Turn each root into a factor and write the equation in root form. If no point is given, any a works, so choose a = 1. If a point is given, use it to find a.

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Key Ideas Two Integer Roots Fractional Roots Double Roots Example 2 Roots and a Point Roots and a Point from the Graph

any quadratic, choose a=1 for easy math!

What are the Factors
The parabola opens up. The roots are x = −1 and x = 5.
−3−2123467−10−8−6−4−22−15

If x = −1

x=−1
+ 1+ 1

Add 1 to both sides.

x + 1 = 0

so (x + 1) is a factor.

If x = 5

x=5
− 5− 5

Subtract 5 from both sides.

x − 5 = 0

so (x − 5) is a factor.

If the problem doesn't tell you a point the graph goes through, then a can be anything. Choose 1 to make the problem easy.

y = 1(x + 1)(x − 5)
Easy

Two Integer Roots

No point is given, so choose a = 1.

Example
Write any quadratic equation that passes through the roots 3 and −6 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.

If x = −6

x=−6
+ 6+ 6

Add 6 to both sides.

x + 6 = 0

so (x + 6) is a factor.

If x = 3

x=3
− 3− 3

Subtract 3 from both sides.

x − 3 = 0

so (x − 3) is a factor.

If the problem doesn't tell you a point the graph goes through, then a can be anything. Choose 1 to make the problem easy.

y = 1(x + 6)(x − 3)

Answer: y = 1(x + 6)(x − 3)

1
Write any quadratic equation that passes through the roots −4 and 2 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −4

x=−4
+ 4+ 4

Add 4 to both sides.

x + 4 = 0

so (x + 4) is a factor.

If x = 2

x=2
− 2− 2

Subtract 2 from both sides.

x − 2 = 0

so (x − 2) is a factor.

No point is given, so choose a = 1.

y = 1(x + 4)(x − 2)

Answer: y = 1(x + 4)(x − 2)

2
Write any quadratic equation that passes through the roots 8 and −3 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −3

x=−3
+ 3+ 3

Add 3 to both sides.

x + 3 = 0

so (x + 3) is a factor.

If x = 8

x=8
− 8− 8

Subtract 8 from both sides.

x − 8 = 0

so (x − 8) is a factor.

No point is given, so choose a = 1.

y = 1(x + 3)(x − 8)

Answer: y = 1(x + 3)(x − 8)

3
Write any quadratic equation that passes through the roots −7 and −2 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −7

x=−7
+ 7+ 7

Add 7 to both sides.

x + 7 = 0

so (x + 7) is a factor.

If x = −2

x=−2
+ 2+ 2

Add 2 to both sides.

x + 2 = 0

so (x + 2) is a factor.

No point is given, so choose a = 1.

y = 1(x + 7)(x + 2)

Answer: y = 1(x + 7)(x + 2)

4
Write any quadratic equation that passes through the roots 5 and 6 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = 5

x=5
− 5− 5

Subtract 5 from both sides.

x − 5 = 0

so (x − 5) is a factor.

If x = 6

x=6
− 6− 6

Subtract 6 from both sides.

x − 6 = 0

so (x − 6) is a factor.

No point is given, so choose a = 1.

y = 1(x − 5)(x − 6)

Answer: y = 1(x − 5)(x − 6)

5
Write any quadratic equation that passes through the roots 0 and 9 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = 0

x = 0

so x is a factor.

If x = 9

x=9
− 9− 9

Subtract 9 from both sides.

x − 9 = 0

so (x − 9) is a factor.

No point is given, so choose a = 1.

y = 1(x)(x − 9)

Answer: y = 1(x)(x − 9)

Medium

Fractional Roots

Clear each denominator, then choose a = 1.

Example
Write any quadratic equation that passes through the roots 23 and −4 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.

If x = −4

x=−4
+ 4+ 4

Add 4 to both sides.

x + 4 = 0

so (x + 4) is a factor.

If x = 23

x=23
× 3× 3

Multiply both sides by 3.

3x = 2

3x=2
− 2− 2

Subtract 2 from both sides.

3x − 2 = 0

so (3x − 2) is a factor.

If the problem doesn't tell you a point the graph goes through, then a can be anything. Choose 1 to make the problem easy.

y = 1(x + 4)(3x − 2)

Answer: y = 1(x + 4)(3x − 2)

1
Write any quadratic equation that passes through the roots −12 and 6 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −12

x=−12
× 2× 2

Multiply both sides by 2.

2x = −1

2x=−1
+ 1+ 1

Add 1 to both sides.

2x + 1 = 0

so (2x + 1) is a factor.

If x = 6

x=6
− 6− 6

Subtract 6 from both sides.

x − 6 = 0

so (x − 6) is a factor.

No point is given, so choose a = 1.

y = 1(2x + 1)(x − 6)

Answer: y = 1(2x + 1)(x − 6)

2
Write any quadratic equation that passes through the roots 35 and −2 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −2

x=−2
+ 2+ 2

Add 2 to both sides.

x + 2 = 0

so (x + 2) is a factor.

If x = 35

x=35
× 5× 5

Multiply both sides by 5.

5x = 3

5x=3
− 3− 3

Subtract 3 from both sides.

5x − 3 = 0

so (5x − 3) is a factor.

No point is given, so choose a = 1.

y = 1(x + 2)(5x − 3)

Answer: y = 1(x + 2)(5x − 3)

3
Write any quadratic equation that passes through the roots −43 and 1 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −43

x=−43
× 3× 3

Multiply both sides by 3.

3x = −4

3x=−4
+ 4+ 4

Add 4 to both sides.

3x + 4 = 0

so (3x + 4) is a factor.

If x = 1

x=1
− 1− 1

Subtract 1 from both sides.

x − 1 = 0

so (x − 1) is a factor.

No point is given, so choose a = 1.

y = 1(3x + 4)(x − 1)

Answer: y = 1(3x + 4)(x − 1)

4
Write any quadratic equation that passes through the roots 12 and 25 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = 25

x=25
× 5× 5

Multiply both sides by 5.

5x = 2

5x=2
− 2− 2

Subtract 2 from both sides.

5x − 2 = 0

so (5x − 2) is a factor.

If x = 12

x=12
× 2× 2

Multiply both sides by 2.

2x = 1

2x=1
− 1− 1

Subtract 1 from both sides.

2x − 1 = 0

so (2x − 1) is a factor.

No point is given, so choose a = 1.

y = 1(5x − 2)(2x − 1)

Answer: y = 1(5x − 2)(2x − 1)

5
Write any quadratic equation that passes through the roots −34 and 13 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

If x = −34

x=−34
× 4× 4

Multiply both sides by 4.

4x = −3

4x=−3
+ 3+ 3

Add 3 to both sides.

4x + 3 = 0

so (4x + 3) is a factor.

If x = 13

x=13
× 3× 3

Multiply both sides by 3.

3x = 1

3x=1
− 1− 1

Subtract 1 from both sides.

3x − 1 = 0

so (3x − 1) is a factor.

No point is given, so choose a = 1.

y = 1(4x + 3)(3x − 1)

Answer: y = 1(4x + 3)(3x − 1)

Hard 1

Double Roots

One root is listed. Use that factor twice.

Example
Write any quadratic equation that passes through the root 4 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.

Only one root is listed, so it is a double root.

If x = 4

x=4
− 4− 4

Subtract 4 from both sides.

x − 4 = 0

so (x − 4) is a factor.

Use the factor twice.

If the problem doesn't tell you a point the graph goes through, then a can be anything. Choose 1 to make the problem easy.

y = 1(x − 4)²

Answer: y = 1(x − 4)²

1
Write any quadratic equation that passes through the root −3 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

Only one root is listed, so it is a double root.

If x = −3

x=−3
+ 3+ 3

Add 3 to both sides.

x + 3 = 0

so (x + 3) is a factor.

Use the factor twice.

No point is given, so choose a = 1.

y = 1(x + 3)²

Answer: y = 1(x + 3)²

2
Write any quadratic equation that passes through the root 7 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

Only one root is listed, so it is a double root.

If x = 7

x=7
− 7− 7

Subtract 7 from both sides.

x − 7 = 0

so (x − 7) is a factor.

Use the factor twice.

No point is given, so choose a = 1.

y = 1(x − 7)²

Answer: y = 1(x − 7)²

3
Write any quadratic equation that passes through the root −5 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

Only one root is listed, so it is a double root.

If x = −5

x=−5
+ 5+ 5

Add 5 to both sides.

x + 5 = 0

so (x + 5) is a factor.

Use the factor twice.

No point is given, so choose a = 1.

y = 1(x + 5)²

Answer: y = 1(x + 5)²

4
Write any quadratic equation that passes through the root 23 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

Only one root is listed, so it is a double root.

If x = 23

x=23
× 3× 3

Multiply both sides by 3.

3x = 2

3x=2
− 2− 2

Subtract 2 from both sides.

3x − 2 = 0

so (3x − 2) is a factor.

Use the factor twice.

No point is given, so choose a = 1.

y = 1(3x − 2)²

Answer: y = 1(3x − 2)²

5
Write any quadratic equation that passes through the root −34 in root form y = a(x − r)(x − r). Note, there is more than one correct answer depending on your value of a.
Show Answer

Only one root is listed, so it is a double root.

If x = −34

x=−34
× 4× 4

Multiply both sides by 4.

4x = −3

4x=−3
+ 3+ 3

Add 3 to both sides.

4x + 3 = 0

so (4x + 3) is a factor.

Use the factor twice.

No point is given, so choose a = 1.

y = 1(4x + 3)²

Answer: y = 1(4x + 3)²

Example 2
Write a quadratic equation with roots −2 and 4 that passes through (0, −4). Write your answer in root form y = a(x − r)(x − r).

x = −2, so (x + 2) is a factor.

x = 4, so (x − 4) is a factor.

y = a(x − r)(x − s)

y = a(x + 2)(x − 4)

Substitute (0, −4).

−4 = a(0 + 2)(0 − 4)

−4 = a(2)(−4)

−4 = −8a

Divide both sides by −8.

a = −4−8

a = 12

y = 12(x + 2)(x − 4)
Hard 2

Roots and a Point

The roots and a point are given. The point tells you a. Answer in root form.

1
Write a quadratic equation with roots 1 and −5 that passes through (0, −10). Write your answer in root form y = a(x − r)(x − r).
Show Answer

x = −5, so (x + 5) is a factor.

x = 1, so (x − 1) is a factor.

y = a(x − r)(x − s)

y = a(x + 5)(x − 1)

Substitute (0, −10).

−10 = a(0 + 5)(0 − 1)

−10 = a(5)(−1)

−10 = −5a

Divide both sides by −5.

a = −10−5

a = 2

y = 2(x + 5)(x − 1)

Answer: y = 2(x + 5)(x − 1)

2
Write a quadratic equation with roots 3 and 6 that passes through (4, 2). Write your answer in root form y = a(x − r)(x − r).
Show Answer

x = 3, so (x − 3) is a factor.

x = 6, so (x − 6) is a factor.

y = a(x − r)(x − s)

y = a(x − 3)(x − 6)

Substitute (4, 2).

2 = a(4 − 3)(4 − 6)

2 = a(1)(−2)

2 = −2a

Divide both sides by −2.

a = 2−2

a = −1

y = −1(x − 3)(x − 6)

Answer: y = −1(x − 3)(x − 6)

3
Write a quadratic equation with roots −3 and 3 that passes through (1, −4). Write your answer in root form y = a(x − r)(x − r).
Show Answer

x = −3, so (x + 3) is a factor.

x = 3, so (x − 3) is a factor.

y = a(x − r)(x − s)

y = a(x + 3)(x − 3)

Substitute (1, −4).

−4 = a(1 + 3)(1 − 3)

−4 = a(4)(−2)

−4 = −8a

Divide both sides by −8.

a = −4−8

a = 12

y = 12(x + 3)(x − 3)

Answer: y = 12(x + 3)(x − 3)

4
Write a quadratic equation with roots 0 and −6 that passes through (−2, −12). Write your answer in root form y = a(x − r)(x − r).
Show Answer

x = −6, so (x + 6) is a factor.

x = 0, so x is a factor.

y = a(x − r)(x − s)

y = a(x + 6)(x)

Substitute (−2, −12).

−12 = a(−2 + 6)(−2)

−12 = a(4)(−2)

−12 = −8a

Divide both sides by −8.

a = −12−8

a = 32

y = 32(x + 6)(x)

Answer: y = 32(x + 6)(x)

5
Write a quadratic equation with roots 1 and −1 that passes through (3, 6). Write your answer in root form y = a(x − r)(x − r).
Show Answer

x = −1, so (x + 1) is a factor.

x = 1, so (x − 1) is a factor.

y = a(x − r)(x − s)

y = a(x + 1)(x − 1)

Substitute (3, 6).

6 = a(3 + 1)(3 − 1)

6 = a(4)(2)

6 = 8a

Divide both sides by 8.

a = 68

a = 34

y = 34(x + 1)(x − 1)

Answer: y = 34(x + 1)(x − 1)

Hard 3

Roots and a Point from the Graph

Read the roots from the x-intercepts. Use the labeled point, or choose a marked point that is not a root. Answer in root form.

1
Write a quadratic equation for this parabola in root form y = a(x − r)(x − r). Use the roots and the labeled point.
−4−3−1134−7−4−3−2−11234−22(0, −4)
Show Answer

The x-intercepts are the roots.

x = −2, so (x + 2) is a factor.

x = 2, so (x − 2) is a factor.

The labeled point is (0, −4).

y = a(x − r)(x − s)

y = a(x + 2)(x − 2)

Substitute (0, −4).

−4 = a(0 + 2)(0 − 2)

−4 = a(2)(−2)

−4 = −4a

Divide both sides by −4.

a = −4−4

a = 1

y = 1(x + 2)(x − 2)

Answer: y = 1(x + 2)(x − 2)

2
Write a quadratic equation for this parabola in root form y = a(x − r)(x − r). Use the roots and the labeled point.
−2−112356−2−11234567804(2, 4)
Show Answer

The x-intercepts are the roots.

x = 0, so x is a factor.

x = 4, so (x − 4) is a factor.

The labeled point is (2, 4).

y = a(x − r)(x − s)

y = a(x)(x − 4)

Substitute (2, 4).

4 = a(2)(2 − 4)

4 = a(2)(−2)

4 = −4a

Divide both sides by −4.

a = 4−4

a = −1

y = −1(x)(x − 4)

Answer: y = −1(x)(x − 4)

3
Write a quadratic equation for this parabola in root form y = a(x − r)(x − r). Use the roots and one marked point that is not a root.
−3−21245−6−5−4−3−212345−13
Show Answer

The x-intercepts are the roots.

x = −1, so (x + 1) is a factor.

x = 3, so (x − 3) is a factor.

Use (1, −4).

(0, −3) gives the same a.

y = a(x − r)(x − s)

y = a(x + 1)(x − 3)

Substitute (1, −4).

−4 = a(1 + 1)(1 − 3)

−4 = a(2)(−2)

−4 = −4a

Divide both sides by −4.

a = −4−4

a = 1

y = 1(x + 1)(x − 3)

Answer: y = 1(x + 1)(x − 3)

4
Write a quadratic equation for this parabola in root form y = a(x − r)(x − r). Use the roots and one marked point that is not a root.
−6−5−3−2−112−6−5−4−3−2−112345−40
Show Answer

The x-intercepts are the roots.

x = −4, so (x + 4) is a factor.

x = 0, so x is a factor.

Use (−2, −4).

(−3, −3) gives the same a.

y = a(x − r)(x − s)

y = a(x + 4)(x)

Substitute (−2, −4).

−4 = a(−2 + 4)(−2)

−4 = a(2)(−2)

−4 = −4a

Divide both sides by −4.

a = −4−4

a = 1

y = 1(x + 4)(x)

Answer: y = 1(x + 4)(x)

5
Write a quadratic equation for this parabola in root form y = a(x − r)(x − r). Use the roots and one marked point that is not a root.
−11345789−4−2246826
Show Answer

The x-intercepts are the roots.

x = 2, so (x − 2) is a factor.

x = 6, so (x − 6) is a factor.

Use (4, −2).

(0, 6) gives the same a.

y = a(x − r)(x − s)

y = a(x − 2)(x − 6)

Substitute (4, −2).

−2 = a(4 − 2)(4 − 6)

−2 = a(2)(−2)

−2 = −4a

Divide both sides by −4.

a = −2−4

a = 12

y = 12(x − 2)(x − 6)

Answer: y = 12(x − 2)(x − 6)

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